Determination of Crystal Structures by X-ray Diffraction

1. Lec. 6,7
1

2. • Diffraction gratings must have spacings comparable to the
wavelength of diffracted radiation.
• Can’t resolve spacings  
• Spacing is the distance between parallel planes of atoms.
2

3. X-Ray Diffractıon Methods
Von Laue Rotating Crystal Powder
Orientation
Single Crystal
Polychromatic
Beam, Fixed
Angle single 
Lattice constant
Single Crystal
Monchromatic
Beam, Variable
Angle 
Varied by
rotation
Lattice
Parameters
Poly Crystal
Monchromatic
Beam, Variable
Angle Many s
(orientations)
3

4. Laue Method
• The Laue method is mainly used to determine the
orientation of large single crystals while radiation is
reflected from, or transmitted through a fixed crystal.
• The diffracted beams form arrays of
spots, that lie on curves on the film.
• The Bragg angle is fixed
for every set of planes in the
crystal. Each set of planes picks
out & diffracts the particular
wavelength from the white radiation
that satisfies the Bragg law for
the values of d & θ involved.
4

5. Transmission Laue Method
• In the transmission Laue method, the film is
placed behind the crystal to record beams which
are transmitted through the crystal.
X-Rays
Film
Single
Crystal
• In the transmission Laue method,
the film is placed behind the
crystal to record beams which are
transmitted through the crystal.
• One side of the cone of Laue
reflections is defined by the
transmitted beam.
• The film intersects the cone,
with the diffraction spots
generally lying on an ellipse. 5

6. Crystal Structure Determination
by the Laue Method
• The Laue method is mainly used to determine the
crystal orientation.
• Although the Laue method can also be used to
determine the crystal structure, several
wavelengths can reflect in different orders
from the same set of planes, with the different
order reflections superimposed on the same spot
in the film. This makes crystal structure
determination by spot intensity diffucult.
• The rotating crystal method overcomes this problem.
6

7. Rotatıng Crystal Method
• In the rotating crystal
method, a single crystal
is mounted with an axis
normal to a
monochromatic x-ray
beam. A cylindrical film
is placed around it & the
crystal is rotated about
the chosen axis.
• As the crystal rotates, Sets of lattice planes will at some
point make the correct Bragg angle
for the monochromatic incident beam, & at that point a diffracted
beam will be formed.
7

8. Rotatıng Crystal Method
The Lattice constant of the crystal can
be determined with this method. For a
given wavelength λ if the angle θ at
which a reflection occurs is known, d can
be determined by using Bragg’s Law.
a
2 2 2
d
h k l

 
2d sin  n
8

9. Rotatıng Crystal Method
The reflected beams are located on the surfaces
of imaginary cones. By recording the diffraction
patterns (both angles & intensities) for various
crystal orientations, one can determine the shape
& size of unit cell as well as the arrangement of
atoms inside the cell.
Film
9

10.  For electromagnetic radiation to be diffracted the spacing
in the grating should be of the same order as the wavelength
 In crystals the typical interatomic spacing ~ 2-3 Å so the
suitable radiation is X-rays
 Hence, X-rays can be used for the study of crystal structures
Beam of electrons Target
X-rays
An accelerating (/decelerating) charge radiates electromagnetic radiation
10

11. Relationship of the Bragg angle (θ) and the experimentally
measured diffraction angle (2θ).
X-ray
intensity
(from
detector)

c
d 
n
2 sin c
11

12. Mo Target impacted by electrons accelerated by a 35 kV potential
Intensity
0.2 0.6 1.0 1.4
Wavelength ()
White
radiation
Characteristic radiation →
due to energy transitions
in the atom
K
K
12

13. Target Metal  Of K radiation (Å)
Mo 0.71
Cu 1.54
Co 1.79
Fe 1.94
Cr 2.29
13

14. BRAGG’s EQUATION
d




Ray 1
Ray 2

Deviation = 2
 The path difference between ray 1 and ray 2 = 2d Sin
 For constructive interference: n = 2d Sin
14

15. 15

16. 16

17. Note that in the Bragg’s equation:
 The interatomic spacing (a) along the plane does not appear
 Only the interplanar spacing (d) appears
 Change in position or spacing of atoms along the plane should not affect
Bragg’s condition !!
d
Note: shift (systematic) is
actually not a problem!
17

18.  Bragg’s equation is a negative law
 If Bragg’s eq. is NOT satisfied  NO reflection can occur
 If Bragg’s eq. is satisfied  reflection MAY occur
 Diffraction = Reinforced Coherent Scattering
Reflection versus Scattering
Reflection Diffraction
Occurs from surface Occurs throughout the bulk
Takes place at any angle Takes place only at Bragg angles
~100 % of the intensity may be reflected Small fraction of intensity is diffracted
X-rays can be reflected at very small angles of incidence
18

19.  n = 2d Sin, n= 1, 2, 3, …
 n is an integer and is the order of the reflection
 For Cu K radiation ( = 1.54 Å) and d110= 2.22 Å
n Sin 
1 0.34 20.7º First order reflection from (110)
2 0.69 43.92º
Second order reflection from (110)
Also written as (220)
a
2 2 2 h k l
dhkl
 

8
220
a
d 
2
110
a
d 
1
2
220 
d
110
d
19

20. In XRD nth order reflection from (h k l) is considered as 1st order reflection
from (nh nk nl)
 2 sin hkl n  d
dhkl 
 2 sin
n
 2 sin nh nk nl  d
20

21. 21

22. The Powder Method
• If a powdered crystal is used instead of a single
crystal, then there is no need to rotate it, because
there will always be some small crystals at an
orientation for which diffraction is permitted.
Here a monochromatic X-ray beam is incident on
a powdered or polycrystalline sample.
• Useful for samples that are difficult to obtain in
single crystal form.
• The powder method is used to determine the
lattice parameters accurately. Lattice parameters
are the magnitudes of the primitive vectors a, b
and c which define the unit cell for the crystal.
22

23. The Powder Method
• For every set of crystal planes, by chance,
one or more crystals will be in the correct
orientation to give the correct Bragg angle
to satisfy Bragg's equation. Every crystal
plane is thus capable of diffraction.
• Each diffraction line is made up of a large
number of small spots, each from a separate
crystal. Each spot is so small as to give the
appearance of a continuous line.
23

24. The Powder Method
• If a monochromatic X-ray beam is directed
at a single crystal, then only one or two
diffracted beams may result. See figure
• For a sample of several randomly orientated
single crystals, the diffracted beams will lie
on the surface of several cones. The cones
may emerge in all directions, forwards and
backwards. See figure
• For a sample of hundreds of crystals
(powdered sample), the diffracted beams
form continuous cones. A circle of film is
used to record the diffraction pattern as
shown. Each cone intersects the film giving
diffraction lines. The lines are seen as arcs
on the film. See figure
24

25. THE POWDER METHOD
Cone of diffracted rays
25

26. POWDER METHOD
Diffraction cones and the Debye-Scherrer geometry
Different cones for different reflections
Film may be replaced with detector
26

27. Debye Scherrer Camera
• A small amount of powdered material is sealed into a fine
capillary tube made from glass that does not diffract X-Rays.
• The sample is placed in the Debye Scherrer camera and
is accurately aligned to be in the center of the camera. X-Rays
enter the camera through a collimator.
• The powder diffracts the X-Rays
in accordance with Braggs Law to
produce cones of diffracted
beams. These cones intersect a
strip of photographic film located
in the cylindrical camera to
produce a characteristic set of
arcs on the film.
27

28. Powder Diffraction Film
• When the film is removed from the
camera, flattened & processed, it shows
the diffraction lines & the holes for the
incident & transmitted beams.
28

29. Some Typical Measurement Results
• Laue - “white” X-rays
– Yields stereoscopic projection of reciprocal lattice
• Rotating-Crystal method: monochromatic X-rays
– Fix source & rotate crystal to reveal reciprocal lattice
• Powder diffraction - monochromatic X-rays
– Powder sample to reveal “all” directions of RL
1
0.5
0
Ce
Y
0.8
CoIn
0.2
5
I/Imax
CeCoIn5 Theory
20 30 40 50 60 70 80 90
Normalized Counts
2
29

30. Photograph of a
XRD Diffractometer
(Courtesy H&M Services.)
30

31. (a) Diagram of a
diffractometer
showing a powdered
sample, incident &
diffracted beams.
(b) Diffraction Pattern
from a sample of
gold powder.
31

32. Example (From the Internet)
The results of a diffraction experiment using X-Rays
with λ = 0.7107 Å (radiation obtained from a
molybdenum, Mo, target) show that diffracted peaks
occur at the following 2θ angles:
Find: The crystal structure, the indices of the plane
producing each peak, & the lattice parameter of the
material. 32

33. Example (Solution)
First calculate the sin2 θ value for each peak, then
divide through by the lowest denominator, 0.0308.
33

34. Example (Solution Continued)
Then use the 2θ values for any of the peaks to
calculate the interplanar spacing & thus the lattice
parameter.
Picking Peak 8: 2θ = 59.42° or θ = 29.71°
Ǻ
So, for example:
0.71699
  
2 sin(29.71)
a d h k l
    
Ǻ
(0.71699)(4) 2.868
0.7107
2 sin
2 2 2
400
0 400
d


This is the lattice parameter for body-centered cubic iron.
34

35. Applications of XRD
Note: XRD is a nondestructive technique!
Some uses of XRD include:
1. Distinguishing between crystalline & amorphous
materials.
2. Determination of the structure of crystalline
materials.
3. Determination of electron distribution within the
atoms, & throughout the unit cell.
4. Determination of the orientation of single crystals.
5. Determination of the texture of polygrained
materials.
6. Measurement of strain and small grain size…..etc.
35

36. Advantages & Disadvantages of XRD
Compared to Other Methods
Advantages
• X-Rays are the least expensive, the most
convenient & the most widely used method to
determine crystal structures.
• X-Rays are not absorbed very much by air, so the
sample need not be in an evacuated chamber.
Disadvantages
• X-Rays do not interact very strongly with lighter
elements.
36

37. Diffraction Methods
X-Rays
λ ~ 1 Ǻ
E ~ 104 eV
interact with
electrons,
penetrating
Neutrons
λ ~ 1 Ǻ
E ~ 0.08 eV
interact with
nuclei, highly
penetrating
Electrons
λ ~ 1 Ǻ
E ~ 150 eV
interact with
electrons, less
penetrating
37